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Gâteaux derivative

The derivative of a functional or a mapping which — together with the Fréchet derivative (the strong derivative) — is most frequently used in infinite-dimensional analysis. The Gâteaux derivative at a point $x_0$ of a mapping $f:X\to Y$ from a linear topological space $X$ into a linear topological space $Y$ is the continuous linear mapping $f'_G(x_0):X\to Y$ that satisfies the condition
\begin{equation*}
f(x_0 + h) = f(x_0)+f'_G(x_0)h + \epsilon(h),
\end{equation*}

where $\epsilon(th)/ t \to 0$ as $t\to 0$ in the topology of $Y$ (see also Gâteaux variation). If the mapping $f$ has a Gâteaux derivative at the point $x_0$, it is called Gâteaux differentiable. The theorem on differentiation of a composite function is usually invalid for the Gâteaux derivative. See also Differentiation of a mapping.